Module 8.
Statistical quality control
Lesson 28
CONTROLCHARTS FOR ATTRIBUTES
28.1
Introduction
In
the previous lesson, we have discussed the control charts for variables. In
spite of wide application of 
and R (s - charts) as a powerful tool of diagnosis
of sources of trouble in a production process, their use is restricted because
of the following limitations:
· These are the charts for
variables i.e. for quality characteristics which can be measured and expressed
in numbers.
· In certain situations they
are impracticable and uneconomical e.g., if the number of measurable
characteristics, each of which could be a possible candidate for and R-chart, is too large, say 30000 or so
then obviously there cannot be many control charts.
As an alternative to and R charts, there are control charts for
attributes which can be used for quality characteristics i.e., (i) which can be observed only as an attribute by
classifying an item as defective or non defective that is conforming to
specifications or not. (ii) which are actually
observed as attributes even though they could be measured as variables. In this
lesson, we will discuss the control charts for attributes.
28.2 Control Charts for Number of Defective and Fraction Defective
When the quality characteristic is an
attribute, and each item is recorded as either defective or non defective, to
judge whether the process is in control, one has to ascertain whether the
population fraction defective P is same for all subgroups. This judgment may be
made either on the number of defective say d in the sample or on the fraction
defective p = d/n in the sample,
where n as before denotes the number of items inspected per subgroup.
28.2.1
Control charts for number of defective
28.2.1.1 Standards given
Assuming
that each random sample is taken with replacements or even if taken without
replacements, is taken from a practically infinite population, we may suppose
that d = np is distributed in the binomial form with E(np) = nP and
P being the same for
all sub groups if and only if the process is in
control. Hence if p' be the
specified standard value of P , The control limits for number of defective
chart will be given as
28.2.1.2 Standards
not given
If
no standard value is specified for p, it will have to be estimated from the
samples themselves. The appropriate estimate for the mean fraction
defective is
The
control limits for number defective chart will then be
Note:
Since
can never be negative. Hence, if LCL,
according to either of the formula comes out to be negative then it is to be
taken as zero.
Example 1: The following table gives the number of bottles broken in a sample of size 25:
|
Sample number |
Number of defectives |
Sample number |
Number of defectives |
Sample number |
Number of defectives |
|
1 |
3 |
10 |
3 |
19 |
3 |
|
2 |
3 |
11 |
2 |
20 |
3 |
|
3 |
2 |
12 |
3 |
21 |
2 |
|
4 |
7 |
13 |
1 |
22 |
2 |
|
5 |
1 |
14 |
9 |
23 |
1 |
|
6 |
8 |
15 |
1 |
24 |
0 |
|
7 |
2 |
16 |
0 |
25 |
1 |
|
8 |
1 |
17 |
4 |
||
|
9 |
0 |
18 |
8 |
Construct
the control chart for number defective. State whether the
process is in a state of control.
Solution :
Here
we have a fixed sample size m=25 for each lot .Prepare the following table :
|
Sample
number |
Number
of defectives (di) |
Fraction
defective (pi) |
Sample
number |
Number
of defectives (di) |
Fraction defective (pi) |
Sample
number |
Number
of defectives (di) |
Fraction
defective (pi) |
|
1 |
3 |
0.12 |
10 |
3 |
0.12 |
19 |
3 |
0.12 |
|
2 |
3 |
0.12 |
11 |
2 |
0.08 |
20 |
3 |
0.12 |
|
3 |
2 |
0.08 |
12 |
3 |
0.12 |
21 |
2 |
0.08 |
|
4 |
7 |
0.28 |
13 |
1 |
0.04 |
22 |
2 |
0.08 |
|
5 |
1 |
0.04 |
14 |
9 |
0.36 |
23 |
1 |
0.04 |
|
6 |
8 |
0.32 |
15 |
1 |
0.04 |
24 |
0 |
0 |
|
7 |
2 |
0.08 |
16 |
0 |
0 |
25 |
1 |
0.04 |
|
8 |
1 |
0.12 |
17 |
4 |
0.16 |
Total |
70 |
2.8 |
|
9 |
0 |
0.04 |
18 |
8 |
0.32 |
Calculate mean
fraction defective as
(due to constant sample size )
The
control limits for number defective chart will then be
(to
be taken as 0 as LCL can’t be negative as stated earlier)
The
control limits for np
-chart and various points are shown in fig. 28.1
Fig.
28.1 Control chart for number defective (np- chart)
Since
,
some of the points i.e. 6th, 14th and 18th are
outside the control limits therefore the process is not in a state of control.
28.2.2
Control Charts for Fraction Defective (p -
chart)
28.2.2.1 Standards given
In
case one constructs a control chart for p instead of, np
one uses the relations
Supposing
p’ is the specified standard for P, the control limit for fraction defective
chart will consist of
LCL = 
CL =
UCL = 
where 
28.2.3 Standards not given
Here the common value
of P will be estimated by 
and the control limits for fraction defective
chart will be
LCL
= 
CL
= 
UCL
= 
Where
Here also since p’can
never be negative. Hence if LCL comes out to be negative, then it is to be
taken as zero.
A ž-chart or np-chart
is advantageous because they may be used even for characters that are observed
as variables. The cost of obtaining data on an attribute is usually less
than that for data on variables. The cost of compiling a ž-chart may also
be less since a ž-chart may be used for any number of characteristics and may
replace pairs of , s, or ,
R charts. In case the sample size is constant, it is immaterial whether
one uses the nž-chart or the ž-chart. If
however the sample size varies, then resulting chart will be highly confusing,
whereas in the Ž-chart the central line will be invariant. It is,
therefore, simpler and preferable to use ž-chart in case the sample size
varies. Instead of computing control limits for each sample size separately,
two sets of limits may be computed based on the minimum and maximum sample
sizes. Action need not be taken for points lying within the inner set of
limits, while action must be taken for points lying beyond the outer
limits. For other points, action should be based on exact control limits.
The confusion in ž-chart (or nž-chart) with varying
control limits can be avoided with some additional computation. For that,
instead of plotting p in the control chart one should plot the standardized
value viz.,
According
as the standard value for ž is specified or not, 
being
the weighted mean of sample proportions with sample sizes as weight. The central
line as well as the control limits becomes invariant with n, since obviously
here
LCL =
-3,
CL =
0,
UCL =3
28.2.4 Interpretations of p-chart
1. If all the sample points fall within the control limits without
exhibiting any specific pattern, the process is said to be in control. In
such a case, the observed variations in the fraction defective are attributed
to the stable pattern of chance causes and the average fraction defective p is
taken as the standard fraction defective P.
2. Points outside
the UCL are termed as high spots. These suggest deterioration in the
quality and should be regularly reported to the Production Engineers. The
reason for such deterioration can be known and removed if the details of
conditions under which data were collected, it may be found, if there was any
change of inspection or inspection standards.
3. Points below
LCL are called low spots. Such points represent a situation showing improvement
in the product quality. However, before taking this improvement for
guarantee it should be investigated if there was any slackness in inspection or
not.
4. When a number
of points fall outside the control limits, a revised estimate of P should be
obtained by eliminating all the points that fall above UCL (it is assumed that
points that fall below LCL are not due to faulty inspection). The standard
fraction defective P should be revised periodically in this way.
Example 2: Table
below gives the results of inspection of nuts used in equipment. The nuts were
packed in 20 lots of 100 each.
|
Lot number |
Number defectives |
Lot number |
Number defectives |
|
1 |
5 |
11 |
4 |
|
2 |
10 |
12 |
7 |
|
3 |
12 |
13 |
8 |
|
4 |
8 |
14 |
2 |
|
5 |
6 |
15 |
3 |
|
6 |
5 |
16 |
4 |
|
7 |
6 |
17 |
5 |
|
8 |
3 |
18 |
8 |
|
9 |
3 |
19 |
6 |
|
10 |
5 |
20 |
10 |
Construct
the control chart for fraction defective. State whether
the process is in a state of control.
Solution :
Here we have a fixed lot size n=100 for each lot .Prepare the following
table :
|
Lot no. |
Lot size (n) |
Number defective (np) |
Fraction defective (p) |
Lot no. |
Lot size (n) |
Number defective (np) |
Fraction defective (p) |
|
1 |
100 |
5 |
0.05 |
11 |
100 |
4 |
0.04 |
|
2 |
100 |
10 |
0.1 |
12 |
100 |
7 |
0.07 |
|
3 |
100 |
12 |
0.12 |
13 |
100 |
8 |
0.08 |
|
4 |
100 |
8 |
0.08 |
14 |
100 |
2 |
0.02 |
|
5 |
100 |
6 |
0.06 |
15 |
100 |
3 |
0.03 |
|
6 |
100 |
5 |
0.05 |
16 |
100 |
4 |
0.04 |
|
7 |
100 |
6 |
0.06 |
17 |
100 |
5 |
0.05 |
|
8 |
100 |
3 |
0.03 |
18 |
100 |
8 |
0.08 |
|
9 |
100 |
3 |
0.03 |
19 |
100 |
6 |
0.06 |
|
10 |
100 |
5 |
0.05 |
20 |
100 |
10 |
0.1 |
|
|
|
|
G. total |
|
2000 |
120 |
1.2 |
The
control limits for fraction defective chart will then be
(to be taken as 0 as LCL can’t be negative as stated earlier)
CL =
The
control limits and various points are shown in fig. 28.2
Fig. 28.2 Control chart for
proportion number defective ( fraction defective
p-chart)
The
control chart shown in fig. 28.2 shows that all the points are falling within control
limits. Hence, the process is in a state of control.
28.3 Control
Chart for Number of Defects Per Unit (C-Chart)
The
and
R control charts may be applied to any quality characteristic that is
measurable. The control chart for p may be applied to the results of any
inspection that accepts or rejects individual items of a product. Thus,
both these types of charts are broadly useful in any SQC programme. The control
chart for number of defects per unit (c-chart) has a much more restricted field
of usefulness. In many manufacturing plants there may be no opportunities
for its economic use even though there are dozens of places where & R charts and p charts can be used
advantageously.
28.3.1 Distinction
between a defect and a defective
A defective is an article that in some way fails to conform to one or more given specifications. Each instance of the article’s lack of conformity to specifications is a defect. Every defective contains one or more defects. The n ž chart which was explained previously, applies to the number of defectives in subgroups of constant size. The c-chart which will now be explained, applies to the number of defects in subgroups of constant size. In most cases, each subgroup for the c-chart consists of a single article; the variable C consists of number of defects observed in one article. However, it is not necessary that the subgroup for C-chart be a single article; it is essential only that the subgroup size be constant in the sense that the different subgroups have substantially equal opportunity for the occurrence of defects.
In many kinds of
manufactured articles, the opportunities for defects are numerous, even though
the chances of a defect occurring in any one spot are small. Whenever
this is true, it is correct as a matter of statistical theory to base control
limits on the assumption that the Poisson distribution is applicable. The
limits on the control chart for C are based on this assumption. Some
representative types of defects to which C chart may be applied are as follows
· C is the number of
defective rivets in an aircraft wing or fuselage.
· C is the number of
breakdowns at weak spots in insulation in a given length of insulated wire
subjected to a specified test voltage.
· C is the number of
surface defects observed in a galvanized sheet or painted, plated, or enameled
surface of a given area or number of seeds on the surface of cheese or Paneer.
· C is the number of
“seeds” (small air pockets) observed in a glass bottle.
· C is the number of
imperfections observed in a bolt of cloth.
· C is the number of
surface defects observed in a roll of coated paper or sheet of photographic
film.
28.3.2
Control limits for C-chart
A
control chart for C aims at detecting any differences, that may exist, among
the Poisson distributions for the different subgroups or, in other words among
the l-values for the subgroups.
28.3.2.1
When standards specified
We
know that for a Poisson Variables C with parameter l
Hence
if a standard value for l say C’ is provided, then the
control chart for C will be given by
CL = C’
28.3.3
Standards not specified
When
no standards are specified then l is estimated from observed C value. Supposing
Ci (i=1, 2,---.m) is the C value for the sample taken from ith subgroup, the appropriate estimate of l will
be 
In
the above case we replace C’ by 
the control limits for C-chart is given by
Note: Since C
can’t be negative. Hence if LCL comes out to be negative, it should be
taken as Zero. The above formulae relate to C-charts with samples of
constant size from all subgroups. In most cases each subgroup sample will
consist of a single article.
Example 3 : The following table gives the number of defects (C) noted at the final inspection of a dairy equipment.
|
Equipment
No. |
No.
of defects |
Equipment
No. |
No.
of defects |
|
1 |
7 |
9 |
20 |
|
2 |
15 |
10 |
11 |
|
3 |
13 |
11 |
22 |
|
4 |
18 |
12 |
15 |
|
5 |
10 |
13 |
8 |
|
6 |
14 |
14 |
24 |
|
7 |
7 |
15 |
14 |
|
8 |
10 |
16 |
8 |
Set up control limits for
C chart and state whether the process is in a state of control for the process.
Solution
: Here m=16
Let us
calculate 
The control limits for C Chart
The control limits and various
points are shown in fig. 28.3
Fig.
28.3 Control chart for number of defects per unit (C-chart)
Since
,
all the points are within control limits therefore the process is in a state of
control.